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arXiv · 2501.19209

Toric ideal of matching polytopes and edge colorings

Kenta Mori·Ryo Motomura·Hidefumi Ohsugi·Akiyoshi Tsuchiya
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Abstract

In the present paper, we investigate the maximal degree of minimal generators of the toric ideal of the matching polytope of a graph. It is known that the toric ideal associated to a bipartite graph is generated by binomials of degree at most $3$. We show that this fact is equivalent to a result in the theory of edge colorings of bipartite multigraphs. Moreover, a characterization of bipartite graphs whose toric ideals are generated by quadratic binomials is given. Finally, we discuss the maximal degree of minimal generators of the toric ideal associated to a general graph and give a conjecture.

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Kenta Mori, Ryo Motomura, Hidefumi Ohsugi, Akiyoshi Tsuchiya. 2025-01-31. Toric ideal of matching polytopes and edge colorings. https://doi.org/10.1112/mtk.70099

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